Let \[\sin (\alpha -\beta )=\frac{5}{13}\] and \[cos(\alpha +\beta )=\frac{3}{5},\] then \[\tan (2\alpha )\] is equal to: (Here \[\alpha ,\beta \in \left( 0,\frac{\pi }{4} \right)\])
Let\[g(x)=\] in \[x\] and \[f(x)=\left( \frac{1-x\cos x}{1+x\cos x} \right)\] then \[\int\limits_{\frac{-\pi }{4}}^{\frac{\pi }{4}}{g(f(x))}dx\] is equal to:
If \[\alpha \] and \[\beta \] are the roots of\[{{x}^{2}}-2x+2=0,\] then find minimum value of \[n\] such that \[{{\left( \frac{\alpha }{\beta } \right)}^{n}}=1:\]
A conducting circular loop of radius 'a' and resistance R is kept on a horizontal plane. A vertical time varying magnetic field \[B=2t\] is switched on at time \[t=0\]. Then:
A)
power generated in the coil at any time t is constant
doneclear
B)
flow of charge per unit time from any section of the coil is constant
doneclear
C)
total charge passed through any section between time\[t=0\] to \[\,t=2\] is \[\,\frac{4\pi {{a}^{2}}}{R}\]
A rigid circular loop of radius r and mass m lies in the x-y plane on a flat table and has a current i flowing in it. At this particular place, the earth's magnetic field is \[\overrightarrow{B}={{B}_{x}}\overset{\hat{\ }}{\mathop{i}}\,+{{B}_{z}}\overset{\hat{\ }}{\mathop{k}}\,\] The value of i so that one edge of the loop lifts from the table 's:
In Young's double slit experiment \[\frac{d}{D}={{10}^{-4}}\](d=distance between slits, D = distance of screen from the slits). At a point P on the screen resulting intensity is equal to the intensity due to individual slit\[{{l}_{0}}\].Then the distance of point P from the central maximum is: \[\left( \lambda =6000A \right)\]
Four equal charges of magnitude q each are placed at four corners of a square with its centre at origin and lying in y-z plane. A fifth charge +Q is moved along x-axis. The electrostatic potential energy (U) varies on x-axis as:
Equations of a stationary and a travelling waves are as follows; \[{{y}_{1}}=a\,\sin \,kx\cos \omega t\] and \[{{y}_{2}}=a\sin (\omega t-kx)\] The phase difference between two points \[{{x}_{1}}=\frac{\pi }{3k}\] and \[{{x}_{2}}=\frac{3\pi }{2k}\], are \[{{\phi }_{1}}\] and \[{{\phi }_{2}}\] respectively for the two waves. The ratio \[\frac{{{\phi }_{1}}}{{{\phi }_{2}}}\] is:
A disc of radius \[0.1\,m\] rolls without sliding on a horizontal surface with a velocity of \[6\,m/s\]. It then ascends a smooth continuous track as shown in figure. The height upto which it will ascend is :\[\left( g=10\,m/{{s}^{2}} \right)\]
A particle moves in the x-y plane with velocity \[\overrightarrow{v}=a\overset{\hat{\ }}{\mathop{i}}\,+bt\overset{\hat{\ }}{\mathop{j}}\,\].At the instant \[t=\frac{a\sqrt{3}}{b}\]the magnitudes of tangential, normal and total accelerations are:
A)
\[\frac{\sqrt{3}}{2}b,\,\,\frac{b}{2}\]and b respectively
In the arrangement shown in figure, coefficient of friction between the two blocks is \[\mu =\frac{1}{2}\] The force of friction acting between the two blocks is:
A uniform sphere of radius R is placed on a rough horizontal surface and given a linear velocity \[{{v}_{0}}\]and angular velocity \[{{\omega }_{0}}\] as shown. The sphere comes to rest after moving some distance to the right. It follows that:
Two particles of equal mass have velocities \[{{\overrightarrow{v}}_{1}}=2\overset{\hat{\ }}{\mathop{i}}\,m/s\] and\[{{\overrightarrow{v}}_{2}}=2\overset{\hat{\ }}{\mathop{j}}\,m/s\]. First particle has an acceleration\[{{\overrightarrow{a}}_{1}}=\,(3\overset{\hat{\ }}{\mathop{i}}\,+\,3\overset{\hat{\ }}{\mathop{j}}\,)\,m/{{s}^{2}}\], while the acceleration of the other particle is zero. The centre of mass of the two particles moves in a:
The maximum kinetic energy of photoelectrons emitted from a surface when photons of energy \[6eV\] fall on it is \[4eV\]. The stopping potential in volts is:
A 100 W bulb \[{{B}_{1}}\], and two 60 W bulbs \[{{B}_{2}}\]and \[{{B}_{3}}\], are connected to a 250 V source, as shown in the figure.
Now, \[{{W}_{1}}\], \[{{W}_{2}}\] and \[{{W}_{3}}\]are the output powers of the bulbs \[{{B}_{1}}\], \[{{B}_{2}}\]and \[{{B}_{3}}\] respectively. Then:
A capacitor is filled with an insulator and a certain potential difference is applied to its plates. The energy stored in the capacitor is U. Now the capacitor is disconnected from the source and the insulator is pulled out of the capacitor. The work performed against the forces of electric field in pulling out the insulator is 4U. Then dielectric constant of the insulator is:
In the circuit shown in figure, a conducting wire HE is moved with a constant speed v towards left. The complete circuit is placed in a uniform magnetic field\[\overrightarrow{B}\] perpendicular to the plane of circuit inwards.
In the formula \[X=3Y{{Z}^{2}}\] X and Z have dimensions of capacitance and magnetic induction respectively. What are the dimensions of Y in MKSQ system?
A uniform disc of radius R lies in x-y plane with its centre at origin. Its moment of inertia about the axis \[x\,=\,2R\] and \[y\,=\,0\] is equal to the moment of inertia about the axis \[y\,=\,\,d\] and \[z\,=\,\,0\], where d is equal to:
Refraction takes place at a concave spherical boundary separating glass air medium. For the image to be real, the object distance \[\left( {{\mu }_{g}}=\frac{3}{2} \right)\]
A)
should be greater than three times the radius of curvature of the refracting surface
doneclear
B)
should be greater than two times the radius of curvature of the refracting surface
doneclear
C)
should be greater than the radius of curvature of the refracting surface
doneclear
D)
is independent of the radius of curvature of the refracting surface
A radioactive substance is being produced at a constant rate of 200 nuclei/s. The decay constant of the substance is\[{{1}^{s-1}}\]. After what time the number of radioactive nuclei will become 100. Initially there are no nuclei present
2 mole, equimolar mixture of \[N{{a}_{2}}{{C}_{2}}{{O}_{4}}\] and \[{{H}_{2}}{{C}_{2}}{{O}_{4}}\] required \[{{V}_{1}}L\] of 0.1 M \[KMn{{O}_{4}}\] in acidic medium for complete oxidation. The same amount of the mixture required \[{{V}_{2}}L\] of 0.2 M \[NaOH\]for neutralization. The ratio of Vi to Vi is -
If \[{{a}_{0}}\] he the radius of first Bohr's orbit of H-atom, the de-Broglie's wavelength of an electron revolving in the second Bohr's orbit will be -
10 mole of ideal gas expand isothermally and reversibly from a pressure of 10 atm to 1 atm at 300 K. What is the largest mass which can lifted through a height of 100 meter?
\[COC{{l}_{2}}\] gas dissociates according to the equation, \[COC{{l}_{2}}(g)\,\,\,\,CO(g)+C{{l}_{2}}(g).\] When heated to 700 K the density of the gas mixture at 1.16 atm and at equilibrium is 1.16 g/litre. The degree of dissociation of \[C{{O}_{2}}\] at 700 K is -
Rate constant \[k=2.303\,\,{{\min }^{-1}}\] for a particular reaction. The initial concentration of the reaction is 1 mol/litre then rate of reaction after 1 minute is
An electrolysis of a oxytungsten complex ion using 1.10 A for 40 min produces 0.838 g of tungsten. What is the charge of tungsten in the materials? (Atomic wt., W= 184)
6.0 g of urea (molecular weight = 60) was dissolved in 9.9 moles of water. If the vapour pressure of pure water is \[{{P}^{o}},\] the vapour pressure of solution is -
A colourless water soluble solid 'X' on heating gives equimolar quantities of Y and Z, Y gives dense white fumes \[HCl\] and Z does so with \[N{{H}_{3}}\] Y gives brown precipitate with Nessler's reagent and Z gives white precipitate with nitrates of \[A{{g}^{+}},\]\[P{{b}^{2+}}\] and \[H{{g}^{+}}\] 'X' is-
Calcium imide on hydrolysis gives gas [B] which on oxidation by bleaching powder gives gas [C]. Gas [C] on reaction with magnesium give compound [D] which on hydrolysis gives again gas [B]. Identify [B], [C] and [D]
At the zoo, Ratan saw a species of primate. He had never even heard of before. He said, it was called a white-faced saki. It had long, dark, spiky far, a long fluffy tail, forward-facing brown eyes, a white forehead, and yellow cheeks. Based on this information, the white-faced saki could not bean:
If there are 120 adenine molecules in a B-DNA double helical structure showing 20 coils, what is the number of pyrimidine nudeotides forming three hydrogen bonds in it?
Most wild plants contain toxins that deter animals from eating them. A scientist discovered that a toxin produced by a certain plant was also toxic to the same plant if it was applied to the roots of the plants. As the first step in finding out why the plant was not normally killed by its own toxin, he fractionated some plant cell and found that the toxin was in the fraction that contained the largest cell organelle. He also found that the toxin was no longer toxic after it was heated. Which of the following statement are consistent with the scientist's observations?
(I) The toxin was stored in the central vacuole.
(II) The toxin can cross the plasma membrane but not the membrane of the organelle in which it is stored.
Patients with HIV are susceptible to a variety of infections because
A)
The virus produces cell surface receptors that bind to pathogens, making it easier for those pathogens to be infective.
doneclear
B)
Synthesising a DNA copy of the viral genome makes a person fell sick.
doneclear
C)
HIV attacks and destroys the T helper cells, which are central to mounting an effective immune response, making those individuals more susceptible to other infections.
doneclear
D)
HIV destroys B cells so that antibodies cannot be made in response to invading pathogens.
Find the magnitude of projection of vector \[2\hat{i}+3\hat{j}+\hat{k},\] on a vector which is perpendicular to the plane containing vectors \[\hat{i}+\hat{j}+\hat{k}\] and \[\hat{i}+2\hat{j}+3\hat{k}.\]
Figure shows a \[2.0\,V\] potentiometer used for the determination of internal resistance of a \[1.5\,V\,\]cell. The balance point of the cell in open circuit is\[75\,cm\]. When a resistor of of \[0.5\Omega \] is used in the external circuit of the cell, the balance point shifts to \[60\,cm\] length of the potentiometer wire. Length of wire AB is \[100\,cm\].
When \[0.5\,W\] is used in the external resistance then terminal voltage of cell is
The convex surface of a thin concavo-convex lens of glass of refractive index 1.5 has a radius of curvature of \[20\,cm\]. The concave surface has a radius of curvature of \[60\,cm\]. The convex side is silvered and placed on a horizontal surface as shown in the figure.
The focal length of the combination has the magnitude
If various elements, i.e., resistance, capacitance and inductance which are in series and having values \[1000\Omega \], \[1\mu F\] and \[2.0H\] respectively. Given emf as, \[V=100\sqrt{2}\sin \,1000\,t\,volts\] Voltage across the inductor is
Two parallel vertical metallic rails AB and CD are separated by 1m. They are connected at the two ends by resistances \[{{R}_{1}}\]and \[{{R}_{2}}\] as shown in figure. Horizontal metallic bar of mass \[0.2\,kg\] slides without friction, vertically down the rails under the action of gravity. There is a uniform horizontal magnetic field of 0.6 T perpendicular to the plane of the rails. It is observed that when the terminal velocity is attained, the power dissipated in \[{{R}_{1}}\]and \[{{R}_{2}}\] are \[0.76\,W\] and \[1.2\,W\] respectively
In the figure \[{{m}_{A}}={{m}_{B}}=1kg\]. Block A is neutral while\[{{q}_{B}}=-1C\].Sizes of A and B are negligible. B is released from rest at a distance \[1.8\,\,m\] from A. Initially spring is neither compressed nor elongated.
Smooth x=0 x=1.8m x-axis
If collision between A and B is perfectly inelastic, what is velocity of combined mass just after collision?
In the figure shown a conducting wire PQ of length \[l\,=\,1\,m\], is moved in a uniform magnetic field \[B\,=4T\] with constant velocity \[v\,=\,2\,m/s\]towards right.
Given: \[R=2\Omega \], \[C=1\,\]\[F\] and \[L=4H\]
Currents through resistance, capacitor and inductor at any time t are\[{{l}_{1}}\],\[{{l}_{2}}\] and \[{{l}_{3}}\]respectively, Current through wire PQ is l. Find the force required to move the wire with the given constant velocity of\[2m/s\] at \[t=2s\]:
The resistance X has temperature coefficient \[{{\alpha }_{1}}\]and from RB \[\left[ 9\Omega \,shown \right]\]has \[{{\alpha }_{2}}\] For shown situation balance point is at 10 cm from left end, if temperature of system increases by \[\Delta T\] due to joule heating than the shift in the balance point is [Assume that only the resistance of X and RB changes due to change in temperature and there is no other effect]
What is the radius of a steel sphere that will float on water with exactly half the sphere submerged? Density of steel is \[7.9\times {{10}^{3}}kg/{{m}^{3}}\]and surface tension of water is \[7\times {{10}^{-2}}N\]
A thin rod of length L and mass M is bent at its midpoint into two halves so that the angle between them is\[90{}^\circ \]. The moment of inertia of the bent rod about an axis passing through the bending point and perpendicular to the plane defined by the two halves of the rod is
Heat is flowing through two cylinderical rods of the same material. The diameters of the rods are in the ratio 1 : 2 and the lengths in the ratio 2 : 1. If the temperature difference between the ends is same, then ratio of the rate of flow of heat through them will be
You are given the following cell at 298 K,
\[Zn\left| \,\begin{matrix}
Z{{n}^{++}}\,(aq.) \\
00.1\,M \\
\end{matrix}\, \right|\,\,\,\left| \,\begin{matrix}
HCl\,(aq.) \\
1.0\,lit \\
\end{matrix}\, \right|\,\,\,\left| \,\begin{matrix}
{{H}_{2}}(g) \\
1.0\,atm \\
\end{matrix}\, \right|Pt\] With \[{{E}_{cell}}=0.701\] and \[E_{z{{n}^{2+}}/Zn}^{0}=-0.76\,\,V.\] Which of the following amounts of NaOH (equivalent weight = 40) will just make the pH of cathodic compartment to be equal to 7. 0:
The maximum radius of an atom which can occupy empty spaces (voids) in a body cantred structure, of an element having atomic radius R, without causing any distortion, can be:
How many moles of sucrose should be dissolved in 500 gms of water so as to get a solution which has a difference of \[104{}^\circ C\] between boiling point and freezing point. \[({{K}_{f}}=1.86\,\,K\,\,kg\,\,mo{{l}^{-1}},\] \[{{K}_{b}}=0.52\,\,K\,\,Kg\,\,mo{{l}^{-1}})\]
For the decomposition of \[{{H}_{2}}{{O}_{2}}\] (aq) it was found that \[{{V}_{{{O}_{2}}}}\] (t = 15 min.) was 100 mL (at \[0{}^\circ C\]and 1 atm) while \[{{V}_{{{O}_{2}}}}\] (maximum) was 200 mL (at \[0{}^\circ C\] and 2 atm) If the same reaction had been followed by the titration method and if \[V_{KMn{{O}_{4}}}^{(CM)}\] (t = 0) had been 40 mL, what would \[V_{KMn{{O}_{4}}}^{(cM)}\](t = 15 min) have been?
Arrangement of gene sequences in representative 50kb segments of two organisms is shown. Genes above the line are transcribed left to right and below the line to the left.
(I) Each segment indicates single stranded. DNA in 5' to 3' orientation (left to right).
(II) Segment I is likely to represent a prokaryote and segment II, a eukaryote such as worm.
(III)Segment II is less likely to have stop codons as compared to segment I.
In humans, attached earlobes are a dominant feature over free earlobes while hypertrichosis of the ear is a holandric (Y-linked) feature. A person with attached earlobes and extensive hair on pinna married another person having free earlobes. The couple had one son with attached earlobes and hairy pinna, another son with free earlobes and hairy pinna and two daughters with attached earlobes. One of the daughters married a person with free earlobes and sparse hair on pinna. They had two sons. What would be the characteristics of their pinnae?
A)
There would be equal chances for both having free or attached earlobes and sparse hair on pinnae.
doneclear
B)
They would have hairy pinnae and there would be 1 in 8 chance that both will have attached earlobes.
doneclear
C)
There is an equal chance for the two to have either hairy pinnae or sparse hair on pinnae.
doneclear
D)
Both will have free earlobes and extensive hair on pinnae.
Composition of various parts of the circulatory system in humans (Artery, vein, arteriole, venule and capillary) is depicted in the form of bar graphs. Select the correct option with appropriate part/structure and composition.
The restriction fragment shown below contains a gene whose recessive allele is lethal. The normal allele has restriction sites for enzymes \[BamI\]at sites I and II whereas recessive allele lacks site I. An individual who has a sister with the lethal trait is being tested to determine if he is a carrier of that lethal trait.
Which of the band patterns would be produced on a gel if he is a carrier?
Many molecules are chiral \[i.e.,\] their mirror images are non-superimposable. Which of the following could be affected if the chirality of a molecule is changed in a biological system?
In corn, three enzymes catalyse the same reaction. Corresponding genes \[\left( {{a}^{+}},{{b}^{+}}and\,{{c}^{+}} \right)\]are located on three different chromosomes. The reaction is as follows: Colourless compound\[\xrightarrow{}\]red compound The normal functioning of any one of these genes is sufficient to convert colourless compound to the red compound. The mutant alleles of \[{{a}^{+}},{{b}^{+}}\] and \[{{\operatorname{c}}^{+}}\] are a, b and c respectively. If red\[\left( {{a}^{+}}/a,{{b}^{+}}/b,{{c}^{+}}/c \right)\] plants are selfed, what proportion of progeny will be colourless?